158 THEORY OF TIDES. 
a spheroidical surface and its inscribed sphere, varies in the 
same proportion as the inclination of the surface, and is to the 
relative force of gravity, depending on that inclination, as 3 
times the density of the shell to 5 times that of the sphere. — 
See Nicholson s Journal, XX. 210. 213. 
fides. Corollary. Hence the ellipticity must be to that which 
would take place if the density n of the sphere were infinite, as 
1 to 1 — — 3 or, in the case of n=5i; nearly as 8 to Q, giving 
for the solar tide *91, and for the lunar 2*263: if n=5, the 
heights are *92 and 2*29 1 respectively 3 if «= 1,2*024 and 
5 042. 
length of the Scholium. The direct attraction, determining the length of 
pendiilum. the p en d u lum in different latitudes, may be calculated in a 
manner nearly similar.— See Nicholsons Journal , XX. 273. 
Theorem H. 
Propagation When the horizontal surface of a liquid is elevated or depress- 
ed a little at a given point, the effect will be propagated in the 
manner of a wave, with a velocity equal to that of a heavy 
body which has fallen through a space equal to half the depth 
of the fluid, the form of the wave remaining similar to that of 
the original elevation or depression. — See Lagrange , Mecanique 
Analitique 3 or Young's Natural Philosophy , II. 63. 
Scholium. In calculating this velocity, it would probably be 
more correct to diminish the multiplier about T tg- or -L, as is 
found to be necessary for determining the velocity of the 
motions of fluids in most other cases. ( See Philosophical 
Transactions , 1808.) 
Theorem I. 
Reflection of A wave of a symmetrical form, with a depression equal and 
waves. similar to its elevation, striking against a solid vertical obstacle, 
will be reflected so as to cause a part of the surface, at the 
distance of one-fourth of its breadth, to remain at rest 3 and if 
there be another opposite obstacle at twice that distance, there 
may be a perpetual vibration between the surfaces, the middle 
point having no vertical motion. — See Young's Natural 
Philosophy , I. 28Q , 777. 
Thus 
